Parser Design for Noncommutative Algebra

Topic: parser.noncommutative_design

Design documentation for MathHook's type-aware LaTeX parser that automatically infers symbol types. Enables seamless support for noncommutative algebra without explicit type annotations in mathematical expressions.

Parser Design for Noncommutative Algebra

Design documentation for MathHook's type-aware LaTeX parser that automatically infers symbol types.

Overview

The parser implements automatic type inference from LaTeX notation, enabling seamless support for noncommutative algebra without explicit type annotations in mathematical expressions.

Key Innovation: LaTeX notation implicitly encodes symbol types:

• \mathbf{A} → Matrix (noncommutative)
• \hat{p} → Operator (noncommutative)
• x → Scalar (commutative, default)

Design Rationale

Why LaTeX Notation?

1. Universal Standard: LaTeX is the de facto standard for mathematical typesetting
2. Rich Semantics: Notation conventions already encode meaning (bold for matrices, hat for operators)
3. User Familiarity: Mathematicians and physicists already know these conventions
4. Minimal Overhead: No separate type annotation syntax needed

Advantages Over Explicit Typing

With Type Inference (our approach):

\mathbf{A}\mathbf{X} = \mathbf{B}

Without Type Inference (alternative):

matrix A * matrix X = matrix B

Benefits:

• Cleaner syntax
• Standard mathematical notation
• No learning curve for users
• Automatic type propagation

Type Inference Rules

Matrix Type

Trigger: \mathbf{identifier}

Rationale: Mathematical convention uses bold for matrices and vectors

Examples:

\mathbf{A}     → Matrix symbol A
\mathbf{B}     → Matrix symbol B
\mathbf{x}     → Matrix symbol x (vector)

Implementation:

• Parser detects \mathbf{...} pattern
• Creates symbol with SymbolType::Matrix
• Preserves identifier name inside braces

Operator Type

Trigger: \hat{identifier}

Rationale: Quantum mechanics convention uses hat notation for operators

Examples:

\hat{p}        → Operator symbol p (momentum)
\hat{x}        → Operator symbol x (position)
\hat{H}        → Operator symbol H (Hamiltonian)

Implementation:

• Parser detects \hat{...} pattern
• Creates symbol with SymbolType::Operator
• Preserves identifier name inside braces

Scalar Type (Default)

Trigger: Plain identifier (no special notation)

Rationale: Standard variables are commutative by default

Examples:

x              → Scalar symbol x
y              → Scalar symbol y
\theta         → Scalar symbol theta

Type System Integration

Symbol Type Enum

#![allow(unused)]
fn main() {
pub enum SymbolType {
    Scalar,      // Commutative (default)
    Matrix,      // Noncommutative
    Operator,    // Noncommutative
    Quaternion,  // Noncommutative
}
}

Commutativity Propagation

Rule: Expression is noncommutative if any operand is noncommutative

Implementation:

#![allow(unused)]
fn main() {
fn commutativity(expr: &Expression) -> Commutativity {
    match expr {
        Expression::Symbol(sym) => sym.commutativity(),
        Expression::Mul(factors) => {
            if factors.iter().any(|f| f.is_noncommutative()) {
                Commutativity::Noncommutative
            } else {
                Commutativity::Commutative
            }
        }
        // ... similar for other operations
    }
}
}

Type Preservation

Guarantee: Symbol types preserved through all operations

Implementation Challenges

Challenge 1: Parser State Management

Problem: LALRPOP is stateless; cannot maintain type context

Solution: Encode types in syntax tree structure (symbol metadata)

Result: Type information flows through AST naturally

Challenge 2: Backward Compatibility

Problem: Existing code assumes all symbols are scalars

Solution: Default to scalar type; noncommutative types opt-in via notation

Result: Zero breaking changes to existing code

Challenge 3: Performance

Problem: Type checking on every operation could be expensive

Solution: Cache type information in symbol itself (O(1) lookup)

Result: No performance regression

Testing Strategy

Unit Tests

Test individual type inference rules:

1. \mathbf{A} → Matrix
2. \hat{p} → Operator
3. x → Scalar
4. Mixed expressions preserve types

Integration Tests

Test end-to-end workflows:

1. Parse → Solve → Format
2. Type preservation through operations
3. Correct LaTeX output formatting

Edge Case Tests

Test boundary conditions:

1. Nested notation
2. Malformed LaTeX
3. Mixed notation precedence
4. Empty identifiers
5. Special characters

Examples

Matrix Type Inference

LaTeX bold notation creates matrix symbols

#![allow(unused)]
fn main() {
use mathhook::parser::latex::parse_latex;

// Bold notation → Matrix symbols
let expr = parse_latex(r"\mathbf{A}\mathbf{B} \neq \mathbf{B}\mathbf{A}")?;

// A and B are noncommutative matrices
// A*B ≠ B*A in general

}

from mathhook.parser import parse_latex

# Bold notation → Matrix symbols
expr = parse_latex(r"\mathbf{A}\mathbf{B} \neq \mathbf{B}\mathbf{A}")

# A and B are noncommutative matrices
# A*B ≠ B*A in general

const { parseLatex } = require('mathhook');

// Bold notation → Matrix symbols
const expr = parseLatex(String.raw`\mathbf{A}\mathbf{B} \neq \mathbf{B}\mathbf{A}`);

// A and B are noncommutative matrices
// A*B ≠ B*A in general

Operator Type Inference

LaTeX hat notation creates operator symbols

#![allow(unused)]
fn main() {
use mathhook::parser::latex::parse_latex;

// Hat notation → Operator symbols
let expr = parse_latex(r"[\hat{x}, \hat{p}] = i\hbar")?;

// Canonical commutation relation
// x and p are noncommutative operators

}

from mathhook.parser import parse_latex

# Hat notation → Operator symbols
expr = parse_latex(r"[\hat{x}, \hat{p}] = i\hbar")

# Canonical commutation relation
# x and p are noncommutative operators

const { parseLatex } = require('mathhook');

// Hat notation → Operator symbols
const expr = parseLatex(String.raw`[\hat{x}, \hat{p}] = i\hbar`);

// Canonical commutation relation
// x and p are noncommutative operators

Mixed Type Expression

Different symbol types in same expression

#![allow(unused)]
fn main() {
use mathhook::parser::latex::parse_latex;

// Quantum mechanics: scalar + operators + matrix
let expr = parse_latex(r"\hbar \omega \hat{a}^\dagger \hat{a} + \mathbf{H}_0")?;

// ℏ and ω: scalars (commutative)
// â†, â: operators (noncommutative)
// H₀: matrix (noncommutative)

}

from mathhook.parser import parse_latex

# Quantum mechanics: scalar + operators + matrix
expr = parse_latex(r"\hbar \omega \hat{a}^\dagger \hat{a} + \mathbf{H}_0")

# ℏ and ω: scalars (commutative)
# â†, â: operators (noncommutative)
# H₀: matrix (noncommutative)

const { parseLatex } = require('mathhook');

// Quantum mechanics: scalar + operators + matrix
const expr = parseLatex(String.raw`\hbar \omega \hat{a}^\dagger \hat{a} + \mathbf{H}_0`);

// ℏ and ω: scalars (commutative)
// â†, â: operators (noncommutative)
// H₀: matrix (noncommutative)

Performance

Time Complexity: O(1) - Type stored in symbol metadata

API Reference

• Rust: mathhook_core::parser::type_inference
• Python: mathhook.parser.type_inference
• JavaScript: mathhook.parser.typeInference

See Also

• parser.latex

• core.symbols

• core.expressions

• advanced.noncommutative_algebra